## Numbers for measurement Chapter 2

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Ashawan  on September 18, 2010

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# Numbers for measurement Chapter 2

 MeasurementThe determination of the demensions, capacity quantity or extent of something.
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#### Definitions

Measurement The determination of the demensions, capacity quantity or extent of something.
Exact number A number that has a value with no uncertainty in it. it is know exactly. (ie. 12 object in a dozen)
Inexact number a number that has a value with a degree of uncertainty in it. result anytie a measurement is made.
Precision an indicator of how close a series of measuremnts on the same object are to each other. involves a series of measurements. Smaller the range between the highest and lowest measurement the more precise.
Accuracy an indicator of how close a measurement(or average of multiple measurements) comes to a true or accepted value
Random error An error originating from uncontrollable variables in an experiment. (ie changes in atmospheric pressure, air currents)
Systemic error An error originating from controllabe variables in an experiment. Constant errors that occur again and again.(ie chipped weight used in a balance would throw off all the measurements in which that weight was used)
Degree of uncertainty Every measurement carries a degree of uncertainty.

Only one estimated digit is ever recorded as part of a measurment.

Most often the uncertainty in the last recorded digit is one unit
Significant figures digits in any measurement that are known with certainty plus one digit that is uncertain. The last digit in a measurement specifies the uncertainty.
Significant Digit RulesRule 1: The digits 1 to 9 inclusive(all of the nonzero digits) always count as significant digits.

Rule 2: Leading Zeros-zeros that occur at the start of a number, that is, zeros that precede all nonzero digits. Such zeros do not count as significant figures.

Rule 3: Confined Zeros are zeros between nonzero digits. Such zeros always count as significant figures.

Rule 4: Trailing zeros are zeros at the end of a number. They are significant if there is a decimal point present in the number of the carry over bars. Otherwise trailing zeros are not significant.
Rounding offThe process of deleting (unwanted) non-significant digits from a calculated number.

Rule 1: If the first digit to be dropped is less than 5, that digit and all the digits that follw it are simply dropped.

Rule 2: If the first digit to be dropped is a digit greater than 5, or a 5 followed by digits other than all zeros, the excess digits are all dropped and the last retained digit is increased in value by one unit

Rule 3: If the first digit to be dropped is a 5 not follwed by any other digit or a 5 followed only by zeros, an odd-even rule applies. Drop the five and any zeros that follow it and then

a) increase the retained digit by one unit if it is odd

b) leave the last retained digit the same if it is even
Multiplying and dividing measurements The answer has the same number of significant digits as the number in the calculation with the fewest significant digits. (significant figures are counted)
Adding and subtracting measurements The uncertainty in the answer should be the same as the number with the greatest uncertainty. (347+ 2.03+23.6)=372.63 yields 373 because "347" has the highest uncertainty of plus or minus 1. So the answer should have an uncertainty of plus or minus 1. (uncertainties are considered)
multiplacation division addition-subtraction. 2 part problem. Use order of operation and apply the rule of significant digit for that operation. Then take that intermediate answer and calculate the next step. then apply the rule of significat digits for that operation
Significant digits in figures where exact numbers are present. The number of significant digits will be based upon the number of significant digits in the measurement.
Scientific notation A system for writing decimal numbers in a more compact form that greatly simplifiesw the mathematical operations of multiplication and division. A x10^n.. A is a number with a single nonzero digit to the left of the decimal.
Uncertainty of a number in Scientific notation Take the uncertainty of the coeffecient(the number on the left from 1 to 9. Then multiply it by exponential term on the Left. EX.
3.753 x 10^4
uncertainty of coeffecient plus or minus 0.001
then multiply 0.001 x 10^4
The true uncertainty is plus or minus 10.
Decimal point movements and Scientific notation Moving the decimal point to the left to obtain a coeffecient from 1 to 9 INCREASES the exponent of the exponential term.

Moving the decimal point to the RIGHT to obtain a coeffecient from 1 to 9 DECREASES the exponent of the exponential term
Multiplying in scientific notation Multiply Coeffecients and apply significant digit rule for multiplication.
Next ADD the exponents of the power terms to generate a new power of ten.
Dividing in scientific notation Divide Coeffecients and apply significant digit rule for Division.

Next SUBTRACT the exponents of the power terms to generate a new power of ten.
Adding and Subtrating in scientific notationExponential term with LIKE exponents:
Add/Subtract coeffecients and follow the rule of significant digits for Adding/Subtracting. Keep the exponential term the same with the same exponent and bring that down in the answer.

Exponential term with Different exponents:
Move the decimal point in one of the coeffecients to get the exponents in the Exponential term to match. (Can be done by factoring the Exponential Term from one Value then combining the non-matching exponential term with the Coeffecient from from that notation.(This moves the decimal )Then Add/Subtract and follow the rule of significant digits for adding and subtracting. Now, the Matching exponential terms are the same as the exponential term in the answer, (just bring it down out of the calculation)

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